Effect of transverse momentum conservation and flow on symmetric cumulants $sc_{2,3} \left \{ 4 \right \}$ and $sc_{2,3,4} \left \{ 6 \right \}$

TL;DR

This work analyzes how transverse momentum conservation (TMC) and collective flow shape symmetric cumulants that couple multiple flow harmonics, focusing on $sc_{2,3}\{4\}$ and $sc_{2,3,4}\{6\}$ and their normalized forms. It first derives pure-TMC expressions for these cumulants, revealing a negative $v_2$–$v_3$ correlation that weakens with multiplicity and increases with momentum, and then extends the framework to include flow up to $v_4$, showing that flow enhances the negative correlations and aligns with ATLAS small-system data for $sc_{2,3}\{4\}$. The paper also provides theoretical predictions for $sc_{2,3,4}\{6\}$ and $nsc_{2,3,4}\{6\}$, highlighting the interplay between TMC and flow and demonstrating that normalized cumulants can expose genuine multi-harmonic correlations in $P(v_m,v_n,\dots,\Psi_m,\dots)$. Overall, the approach offers a framework to disentangle initial-state fluctuations and final-state dynamics in small-system collisions.

Abstract

Symmetric cumulants can improve our understanding of the joint probability distribution function $ P\left ( v_{m},v_{n},v_{k}, \dots,Ψ_{m},Ψ_{n},Ψ_{k},\dots \right )$, potentially offering new insights into the nature of the fluctuations of the quark-gluon plasma produced in relativistic heavy-ion collisions. In this work, the four-particle symmetric cumulants $sc_{2,3} \left \{ 4 \right \}$, six-particle symmetric cumulants $sc_{2,3,4} \left \{ 6 \right \}$, and the normalized cumulants $nsc_{2,3} \left \{ 4 \right \}$ and $nsc_{2,3,4} \left \{ 6 \right \}$ originating from transverse momentum conservation, collective flow, and the interplay between the two effects are calculated. Our results on $sc_{2,3} \left \{ 4 \right \}$ are consistent with the ATLAS data using the subevent cumulant method, facilitating a more profound understanding of the origins of the symmetric cumulant in small systems. Our results on $sc_{2,3,4} \left \{ 6 \right \}$ serve as theoretical predictions for future experimental measurements in small systems.

Figures (4)

• Figure 1: The four-particle symmetric cumulant $sc_{2,3} \left \{ 4 \right \}$ from "pure TMC" (top-left), from "pure TMC+pure flow+interplay" (top-right), and the normalized cumulants $nsc_{2,3} \left \{ 4 \right \}$ from "pure TMC+pure flow+interplay" (bottom-right) as a function of the number of particles $N$ for various values of transverse momenta $p$; and $sc_{2,3} \left \{ 4 \right \}$ from "pure TMC," "pure TMC+pure collective flow," and "pure TMC+pure flow+interplay" as a function of the number of particles $N$ for momentum $p=0.6$ GeV (bottom-left). The ATLAS data for $0.3< p_{T} < 3$ GeV in p+p collisions at 13 TeV using the four-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively data1.
• Figure 2: The six-particle symmetric cumulant $sc_{2,3,4} \left \{ 6 \right \}$ from "pure TMC" (top-left), from "pure TMC+pure flow+interplay" (top-right), normalized cumulants $nsc_{2,3,4} \left \{ 6 \right \}$ from "pure TMC+pure flow+interplay" as a function of the number of particles $N$ for various values of transverse momenta $p$ (bottom-right), and $sc_{2,3,4} \left \{ 6 \right \}$ from "pure TMC," "pure TMC and pure flow," and 'pure TMC+pure flow+interplay" as a function of the number of particles $N$ for momentum $p=0.6$ GeV (bottom-left).
• Figure 3: The symmetric cumulants $sc_{2,3} \left \{ 4 \right \}$ (left column) and $sc_{2,3,4} \left \{ 6 \right \}$ (right column) as a function of the number of particles $N$ for various values of $v_n$ at momentum $p=0.6$ GeV. When varying $v_2$, the values of $v_3$ and $v_4$ are kept constant at 0.006 and 0.0075, respectively. When varying $v_3$, the values of $v_2$ and $v_4$ are kept constant at 0.05 and 0.0075, respectively. When varying $v_4$, the values of $v_2$ and $v_3$ are kept constant at 0.05 and 0.006, respectively. The ATLAS data for $0.3< p_{T} < 3$ GeV in p+p collisions at 13 TeV using the four-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively data1.
• Figure 4: The symmetric cumulants $sc_{2,3} \left \{ 4 \right \}$ (left) and $sc_{2,3,4} \left \{ 6 \right \}$ (right) as a function of the number of particles $N$ for various values of $v_n$ at momentum $p=0.6$ GeV. The ATLAS data for $0.3< p_{T} < 3$ GeV in p+p collisions at 13 TeV using the four-subevent cumulant method are shown for comparisons, where the error bars and boxes represent the statistical and systematic uncertainties, respectively data1.